
## import libraries to ignore warnies
import warnings
warnings.filterwarnings('ignore')


## import libraries

import pandas as pd 
#import numpy as np
import seaborn as sns 
import matplotlib.pyplot as plt
from sklearn.preprocessing import LabelEncoder


#input data
## use filepath/data location

Data = pd.read_csv("/Users/Downloads/WLD_RTFP_country_2023-10-02.csv")


#validate data inputted

Data


#Examing data from csv file noticed no inflation data from Jan-Dec 2007. 


display(Data.loc[(Data['country']=='Nigeria') & (Data['date'] <= '2007-12-01')])


display(Data.loc[(Data['country']=='Nigeria') & (Data['date'] > '2007-12-01')])


# Data consist of various countries, filter the data by selecting Nigeria, dropping unwanted column
#ISO3 =country name Abbrv. and deleting known missing value rows. 

NigData = (Data.loc[(Data['country']=='Nigeria') & 
                    (Data['date']> '2007-12-01')]) 
NigData = NigData.drop(columns=['ISO3','country'])


NigData


# checking the null values in dataset
NigData.isnull().sum()

Open         0
High         0
Low          0
Close        0
Inflation    0
date         0
dtype: int64


# checking the datatypes
NigData.dtypes

Open         float64
High         float64
Low          float64
Close        float64
Inflation    float64
date          object
dtype: object


#convert our date column object to datetime

NigData['date'] = pd.to_datetime(NigData['date'])


# Validating datetime change 
NigData.dtypes

Open                float64
High                float64
Low                 float64
Close               float64
Inflation           float64
date         datetime64[ns]
dtype: object


# checking and drop duplicates if any- No duplucates 

NigData.drop_duplicates()


#Describing data --- Year of min and max inflations

NigData.describe()


#Data manipulation

# date Extracting Components:
NigData['date'] = pd.to_datetime(NigData['date'])
NigData['year'] = NigData['date'].dt.year
NigData['month'] = NigData['date'].dt.month
NigData


#making date an index as this is a time-series data

NigData.set_index('date', inplace=True)


NigData


#Describing data --- Year of min and max inflations

NigData.describe()


#Confirming min inflation value month and year
NigData[NigData['Inflation']  == -22.710000]


#Confirming max inflation value month and year
NigData[NigData['Inflation']  == 77.390000]


#plotting Monthly inflation values by year'

import calendar

ts = NigData['Inflation']

df_plot = ts.groupby([ts.index.month,ts.index.year]).sum().unstack()
df_plot

fig, ax = plt.subplots(figsize=(15,8))
df_plot.plot(ax=ax,legend=True)
plt.title ('Monthly inflation values by year')

# xticks
months = [calendar.month_abbr[i] for i in range(0,13)]
ax.set_xticks(range(13))
ax.set_xticklabels(months)

# plot names in the end
for col in df_plot.columns:
    plt.annotate(col,xy=(plt.xticks()[0][-1]+0.0, df_plot[col].iloc[-1]))


#plotting Monthly close values by year'

to = NigData['Close']

do_plot = to.groupby([to.index.month,ts.index.year]).sum().unstack()
do_plot

fig, ax = plt.subplots(figsize=(15,8))
do_plot.plot(ax=ax,legend=True)  
plt.title ('Monthly close values by year')

# xticks
months = [calendar.month_abbr[i] for i in range(0,13)]
ax.set_xticks(range(13))
ax.set_xticklabels(months)

# plot names in the end
for col in df_plot.columns:
    plt.annotate(col,xy=(plt.xticks()[0][-1]+0.3, do_plot[col].iloc[-1]))


#Plotting inflation index

plt.figure(figsize=(15, 10))
plt.plot(NigData['Close'], label='Close Value', color='orange')
plt.plot(NigData['High'], label='High Value', color='red')
plt.plot(NigData['Low'], label='Low Value', color='green')
plt.plot(NigData['Open'], label='Open Value', color='purple')
plt.plot(NigData['Inflation'], label='Inflation Value', color='Blue')
plt.title('Nigeria Inflation index against Open,Close,High and Low index')
plt.xlabel('Date')
plt.ylabel('Indexes')
plt.legend()
plt.grid(True)
plt.show()


#Box plot for Inflation variable

plt.figure(figsize=(20, 8))
sns.boxplot(x='year',y='Inflation', data=NigData)
plt.title('Box Plot by Year: Inflation')
plt.xlabel('Year')
plt.ylabel('Inflation')
plt.show()


#Plotting Open,Close,High and Low inflation index

plt.figure(figsize=(15, 10))
plt.plot(NigData['Close'], label='Close Value', color='red')
plt.plot(NigData['High'], label='High Value', color='orange')
plt.plot(NigData['Low'], label='Low Value', color='green')
plt.plot(NigData['Open'], label='Open Value', color='purple')
plt.title('Nigeria Open,Close,High and Low inflation index')
plt.xlabel('Date')
plt.ylabel('Indexes')
plt.legend()
plt.grid(True)
plt.show()


#Box plot for close variable

plt.figure(figsize=(20, 8))
sns.boxplot(x='year', y='Close', data= NigData)
plt.title('Box Plot by Year: Close')
plt.xlabel('Year')
plt.ylabel('Close Values')
plt.show()


#Plotting open series

plt.figure(figsize=(14, 8))
plt.plot(NigData['Open'],label='Open')
plt.title('Time Series:open')
plt.xlabel('date')
plt.ylabel('Values')
plt.legend()
plt.show()


#Plotting High series

plt.figure(figsize=(14, 8))
plt.plot(NigData['High'], label='High')
plt.title('Time Series:High')
#plt.xlabel('date')
plt.ylabel('Values')
plt.legend()
plt.show()


#Plotting Low series

plt.figure(figsize=(14, 8))
plt.plot(NigData['Low'], label='low')
plt.title('Time Series:low')
#plt.xlabel('date')
plt.ylabel('Values')
plt.show()


#Plotting Close series
plt.figure(figsize=(14, 8))
plt.plot(NigData['Close'], label='Close')
plt.title('Time Series:close')
plt.xlabel('date')
plt.ylabel('Values')
plt.show()


#Pairplot of all variables

sns.pairplot(NigData[['Open', 'High', 'Low','Close', 'Inflation','year','month']])


plt.show()


# Correlation by Heatmap

plt.figure(figsize=(16, 6))
heatmap = sns.heatmap(NigData.corr(), vmin=-1, vmax=1, annot=True, cmap='coolwarm')
heatmap.set_title('Heatmap showing correlation of Inflation and other variables', fontdict={'fontsize':18}, pad=12);
# save heatmap as .png file
# dpi - sets the resolution of the saved image in dots/inches
# bbox_inches - when set to 'tight' - does not allow the labels to be cropped
plt.savefig('heatmap.png', dpi=300, bbox_inches='tight')


# Correlation by Heatmap

import seaborn as sns

plt.figure(figsize=(8, 12))
heatmap = sns.heatmap(NigData.corr()[['Inflation']].sort_values(by='Inflation', ascending=False), 
                      vmin=-1, vmax=1, annot=True, cmap='BrBG')
heatmap.set_title('Features Correlating with Inflation', fontdict={'fontsize':18}, pad=16)

Text(0.5, 1.0, 'Features Correlating with Inflation')


import xgboost as xgb
from sklearn.model_selection import train_test_split
from sklearn.ensemble import RandomForestRegressor


#splitting test/train to train gradient boosting

x = NigData[['Open', 'High', 'Low','Close','year','month']]
y = NigData['Inflation']
# Split the data into training and testing sets
x_train, x_test, y_train, y_test = train_test_split(x, y,test_size=0.2,random_state=30)


#Train gradient boosting for feature importance

reg = xgb.XGBRegressor(base_score=0.5, booster='gbtree',    
                       n_estimators=1000,
                       early_stopping_rounds=50,
                       objective='reg:linear',
                       max_depth=3,
                       learning_rate=0.01)
reg.fit(x_train, y_train,
        eval_set=[(x_train, y_train), (x_test, y_test)],
        verbose=100)

[0]	validation_0-rmse:27.35638	validation_1-rmse:22.68405
[100]	validation_0-rmse:16.30149	validation_1-rmse:14.72447
[200]	validation_0-rmse:11.54170	validation_1-rmse:11.99255
[300]	validation_0-rmse:9.00370	validation_1-rmse:9.96251
[400]	validation_0-rmse:7.44376	validation_1-rmse:8.88482
[500]	validation_0-rmse:6.43860	validation_1-rmse:8.50751
[600]	validation_0-rmse:5.62716	validation_1-rmse:8.19358
[700]	validation_0-rmse:4.96249	validation_1-rmse:7.90363
[800]	validation_0-rmse:4.38732	validation_1-rmse:7.64217
[900]	validation_0-rmse:4.02986	validation_1-rmse:7.43054
[999]	validation_0-rmse:3.77756	validation_1-rmse:7.33103

XGBRegressor(base_score=0.5, booster='gbtree', callbacks=None,
             colsample_bylevel=None, colsample_bynode=None,
             colsample_bytree=None, device=None, early_stopping_rounds=50,
             enable_categorical=False, eval_metric=None, feature_types=None,
             gamma=None, grow_policy=None, importance_type=None,
             interaction_constraints=None, learning_rate=0.01, max_bin=None,
             max_cat_threshold=None, max_cat_to_onehot=None,
             max_delta_step=None, max_depth=3, max_leaves=None,
             min_child_weight=None, missing=nan, monotone_constraints=None,
             multi_strategy=None, n_estimators=1000, n_jobs=None,
             num_parallel_tree=None, objective='reg:linear', ...)


#feature importance 

fi = pd.DataFrame(data=reg.feature_importances_,
             index=reg.feature_names_in_,
             columns=['importance'])
fi.sort_values('importance').plot(kind='barh', title='Feature Importance')
plt.show()


#Splitting dataframe for close and inflation only

ND = NigData[["Close","Inflation"]]


#validating new df
ND


#plotting inflation and close to see similarities

y1 = ND['Close']
y2 = ND['Inflation']

fig, ax = plt.subplots(nrows=1, ncols=2, figsize=(15, 6))
ax[0].plot(y1)
ax[1].plot(y2)


for ax[0] in ax.flat:
    ax[0].set(xlabel='date')
    ax[0].set(ylabel='Close')
    ax[1].set(ylabel='Inflation')

fig.suptitle('Time Series: Close vs Inflation')
fig.tight_layout()


# Performing granger casuality test 
import numpy as np
from statsmodels.tsa.stattools import grangercausalitytests

def grangers_causation_matrix(ND, variables, test='ssr_chi2test', verbose=False):    
    """Check Granger Causality of all possible combinations of the Time series.
    The rows are the response variable, columns are predictors. The values in the table 
    are the P-Values. P-Values lesser than the significance level (0.05), implies 
    the Null Hypothesis that the coefficients of the corresponding past values is 
    zero, that is, the X does not cause Y can be rejected.
    """
    df = pd.DataFrame(np.zeros((len(variables), len(variables))), columns=variables, index=variables)
    for c in df.columns:
        for r in df.index:
            test_result = grangercausalitytests(ND[[r, c]], maxlag=maxlag, verbose=False)
            p_values = [round(test_result[i+1][0][test][1],4) for i in range(maxlag)]
            if verbose: print(f'Y = {r}, X = {c}, P Values = {p_values}')
            min_p_value = np.min(p_values)
            df.loc[r, c] = min_p_value
    df.columns = [var + '_x' for var in variables]
    df.index = [var + '_y' for var in variables]
    return df


maxlag=12
grangers_causation_matrix(ND, variables = ND.columns)


#decomposing inflation to show trend,seasonality and residue

from pylab import rcParams
import statsmodels.api as sm

rcParams['figure.figsize'] = 18, 8
decomposition = sm.tsa.seasonal_decompose(ND['Inflation'], model='additive')
fig = decomposition.plot()
plt.show()


#defining ADF test for stationarity

from statsmodels.tsa.stattools import adfuller

def check_stationarity(timeseries):
    # Perform the Dickey-Fuller test
    result = adfuller(timeseries, autolag='AIC')
    p_value = result[1]
    print(f'ADF Statistic: {result[0]}')
    print(f'p-value: {p_value}')
    print('Stationary' if p_value < 0.05 else 'Non-Stationary')


#checking ADF test for stationarity for close and inflation

print('Inflation')
print(' ')
check_stationarity(ND['Inflation'])

Inflation
 
ADF Statistic: -3.032539648304694
p-value: 0.031965333762831116
Stationary


# KPSS test
from statsmodels.tsa.stattools import kpss
def kpss_test(NI, **kw):    
    statistic, p_value, n_lags, critical_values = kpss(NI, **kw)
    # Format Output
    print(f'KPSS Statistic: {statistic}')
    print(f'p-value: {p_value}')
    print(f'num lags: {n_lags}')
    print('Critial Values:')
    for key, value in critical_values.items():
        print(f'   {key} : {value}')
    print(f'Result: The series is {"not " if p_value < 0.05 else ""}stationary')

kpss_test(ND['Inflation'])

KPSS Statistic: 0.13151926975925599
p-value: 0.1
num lags: 9
Critial Values:
   10% : 0.347
   5% : 0.463
   2.5% : 0.574
   1% : 0.739
Result: The series is stationary

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/tsa/stattools.py:2022: InterpolationWarning: The test statistic is outside of the range of p-values available in the
look-up table. The actual p-value is greater than the p-value returned.

  warnings.warn(


from statsmodels.graphics.tsaplots import plot_acf, plot_pacf


#autocorrelation plot -q

plt.rc("figure", figsize=(20,10))
plt.figure(figsize=(20,10))
plot_acf(ND['Inflation'])
plt.show()

<Figure size 1440x720 with 0 Axes>


#partial autocorrelation plot - P

plt.rc("figure", figsize=(20,10))
plt.figure(figsize=(20,10))
plot_pacf(ND['Inflation'])
plt.show()

<Figure size 1440x720 with 0 Axes>


#Dropping close variable hence univariate forecasting

NI = ND['Inflation']


#train and test for forecasting. using the last 12 values as test data while other data points for train data

train_data = NI.iloc[:-12]
test_data = NI.iloc[-12:]


#definig index for train data

train_data.index = pd.DatetimeIndex(train_data.index.values,
                               freq=train_data.index.inferred_freq)


train_data.index

DatetimeIndex(['2008-01-01', '2008-02-01', '2008-03-01', '2008-04-01',
               '2008-05-01', '2008-06-01', '2008-07-01', '2008-08-01',
               '2008-09-01', '2008-10-01',
               ...
               '2022-01-01', '2022-02-01', '2022-03-01', '2022-04-01',
               '2022-05-01', '2022-06-01', '2022-07-01', '2022-08-01',
               '2022-09-01', '2022-10-01'],
              dtype='datetime64[ns]', length=178, freq='MS')


#Building ARIMA Model 

from statsmodels.tsa.arima.model import ARIMA


model = ARIMA(train_data,order=(3,0,3))  # 3,0,3 ( arima p d q )

# Training arima model 
Infmodel_fit = model.fit()

print(Infmodel_fit.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                 ARIMA(3, 0, 3)   Log Likelihood                -515.205
Date:                Wed, 01 May 2024   AIC                           1046.410
Time:                        04:14:38   BIC                           1071.864
Sample:                    01-01-2008   HQIC                          1056.732
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         10.3477      4.514      2.292      0.022       1.500      19.195
ar.L1          1.2659      0.213      5.936      0.000       0.848       1.684
ar.L2          0.3277      0.413      0.793      0.428      -0.483       1.138
ar.L3         -0.6231      0.208     -2.991      0.003      -1.031      -0.215
ma.L1          0.1583      0.242      0.654      0.513      -0.316       0.633
ma.L2         -0.7647      0.127     -6.022      0.000      -1.014      -0.516
ma.L3         -0.0817      0.166     -0.493      0.622      -0.407       0.243
sigma2        18.6443      1.473     12.655      0.000      15.757      21.532
===================================================================================
Ljung-Box (L1) (Q):                   0.01   Jarque-Bera (JB):               182.43
Prob(Q):                              0.92   Prob(JB):                         0.00
Heteroskedasticity (H):               0.58   Skew:                            -0.72
Prob(H) (two-sided):                  0.04   Kurtosis:                         7.75
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


# Model evaluation

from sklearn.metrics import mean_absolute_error, mean_squared_error
import numpy as np

fcast= Infmodel_fit.get_forecast(steps=12)
predicted_inf = fcast.predicted_mean

mse = mean_squared_error(test_data, predicted_inf)
rmse = np.sqrt(mse)

print("Mean Squared Error: ", mse)
print("Root Mean Squared Error: ", rmse)

Mean Squared Error:  21.536848806705397
Root Mean Squared Error:  4.640781055674292


#Plotting actual test data and predicted data
plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted_inf, label='Predicted Inflation')
plt.title('ARIMA (3,0,3)')
plt.legend()
plt.show()


fcast.conf_int()


# Print the forecast values : actual v predicted v lower ci v upper ci (95% confidence interval)

result0 = pd.concat([test_data, predicted_inf,fcast.conf_int()], axis=1)
result0 = result0.rename(columns={0: 'Inflation','predicted_mean': 'Pred_Inflation', 'lower Inflation': 'Lower CI',
                                  'upper Inflation': 'Upper CI'})
print(result0)

            Inflation  Pred_Inflation   Lower CI   Upper CI
2022-11-01      13.65        9.599278   1.136343  18.062214
2022-12-01      11.76        7.895341  -6.832552  22.623234
2023-01-01       6.83        7.700740 -11.022703  26.424182
2023-02-01       6.76        6.659408 -15.920565  29.239382
2023-03-01       5.71        6.339140 -19.352330  32.030610
2023-04-01       7.20        5.713670 -22.834247  34.261586
2023-05-01       9.52        5.465780 -25.466755  36.398314
2023-06-01      11.02        5.146540 -27.861379  38.154459
2023-07-01      11.45        5.050907 -29.657317  39.759132
2023-08-01      11.83        4.979680 -31.123331  41.082691
2023-09-01      12.33        5.057097 -32.131452  42.245646
2023-10-01      10.85        5.191350 -32.819296  43.201996


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = result0.index.values


upper = result0['Upper CI']
lower = result0['Lower CI']

result0['Inflation'].plot(color = 'blue', label = 'Actual')
result0['Pred_Inflation'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'grey', label = 'Upper CI')
lower.plot(color = 'grey', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)


plt.show()


#viewing modelprediction for plot 
model303 = model.fit().predict()


#Plotting actual test data and predicted data
plt.plot(NI, label='Actual Inflation')
plt.plot(model303, label='Model predicted Inflation')
plt.plot(predicted_inf, label='Test data Predicted Inflation')
plt.title('ARIMA (3,0,3)')
plt.legend()
plt.show()


# Plot residual errors
residuals = pd.DataFrame(Infmodel_fit .resid)
fig, ax = plt.subplots(1,2)
residuals.plot(title="Residuals", ax=ax[0])
residuals.plot(kind='kde', title='Density', ax=ax[1])
plt.show()


model0 = ARIMA(train_data,order=(2,0,1))
# Training arima modeling 
Infmodel_fit0 = model0.fit()
print(Infmodel_fit0.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                 ARIMA(2, 0, 1)   Log Likelihood                -525.438
Date:                Wed, 01 May 2024   AIC                           1060.877
Time:                        04:15:34   BIC                           1076.786
Sample:                    01-01-2008   HQIC                          1067.328
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         10.7577      6.717      1.602      0.109      -2.407      23.922
ar.L1          1.8537      0.061     30.623      0.000       1.735       1.972
ar.L2         -0.8789      0.059    -14.860      0.000      -0.995      -0.763
ma.L1         -0.6051      0.098     -6.173      0.000      -0.797      -0.413
sigma2        20.9974      1.489     14.097      0.000      18.078      23.917
===================================================================================
Ljung-Box (L1) (Q):                   2.15   Jarque-Bera (JB):                57.22
Prob(Q):                              0.14   Prob(JB):                         0.00
Heteroskedasticity (H):               0.66   Skew:                            -0.33
Prob(H) (two-sided):                  0.11   Kurtosis:                         5.70
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


fcast0= Infmodel_fit0.get_forecast(steps=12)
predicted_inf0 = fcast0.predicted_mean

mse0 = mean_squared_error(test_data, predicted_inf0)
rmse0 = np.sqrt(mse0)

print("Mean Squared Error: ", mse0)
print("Root Mean Squared Error: ", rmse0)

Mean Squared Error:  18.692822022188427
Root Mean Squared Error:  4.323519633607372


plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted_inf0, label='Predicted Inflation')
plt.title ('ARIMA (2,0,1)')
plt.legend()
plt.show()


fcast0.conf_int()


# Print the forecast values : actual v predicted v lower ci v upper ci (95% confidence interval)

result = pd.concat([test_data, predicted_inf0,fcast0.conf_int()], axis=1)
result = result.rename(columns={0: 'Inflation','predicted_mean': 'Pred_Inflation', 'lower Inflation': 'Lower CI',
                                  'upper Inflation': 'Upper CI'})
print(result)

            Inflation  Pred_Inflation   Lower CI   Upper CI
2022-11-01      13.65        9.532591   0.551471  18.513711
2022-12-01      11.76       10.593966  -3.773209  24.961142
2023-01-01       6.83       11.530892  -7.774027  30.835810
2023-02-01       6.76       12.334866 -11.539552  36.209285
2023-03-01       5.71       13.001766 -15.038406  41.041937
2023-04-01       7.20       13.531413 -18.228879  45.291706
2023-05-01       9.52       13.927108 -21.080193  48.934409
2023-06-01      11.02       14.195120 -23.577026  51.967266
2023-07-01      11.45       14.344171 -25.719426  54.407768
2023-08-01      11.83       14.384922 -27.521134  56.290979
2023-09-01      12.33       14.329467 -29.007275  57.666208
2023-10-01      10.85       14.190853 -30.211755  58.593462


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = result.index.values


upper = result['Upper CI']
lower = result['Lower CI']

result['Inflation'].plot(color = 'blue', label = 'Actual')
result['Pred_Inflation'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'grey', label = 'Upper CI')
lower.plot(color = 'grey', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)


plt.show()


#viewing modelprediction for plot 
model201 = model0.fit().predict()


#Plotting actual test data and predicted data
plt.plot(NI, label='Actual Inflation')
plt.plot(model201, label='Model predicted Inflation')
plt.plot(predicted_inf, label='Test data Predicted Inflation')
plt.title('ARIMA(2,0,1)')
plt.legend()
plt.show()


# Plot residual errors
residuals = pd.DataFrame(Infmodel_fit0 .resid)
fig, ax = plt.subplots(1,2)
residuals.plot(title="Residuals", ax=ax[0])
residuals.plot(kind='kde', title='Density', ax=ax[1])
plt.show()


model302 = ARIMA(train_data,order=(3,0,2))
# Training arima modeling 
Infmodel_fit302 = model302.fit()
print(Infmodel_fit302.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                 ARIMA(3, 0, 2)   Log Likelihood                -515.434
Date:                Wed, 01 May 2024   AIC                           1044.868
Time:                        04:15:54   BIC                           1067.140
Sample:                    01-01-2008   HQIC                          1053.900
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         10.5010      5.116      2.053      0.040       0.473      20.529
ar.L1          1.3226      0.121     10.928      0.000       1.085       1.560
ar.L2          0.1994      0.205      0.971      0.332      -0.203       0.602
ar.L3         -0.5530      0.100     -5.526      0.000      -0.749      -0.357
ma.L1          0.0913      0.139      0.656      0.512      -0.182       0.364
ma.L2         -0.7152      0.103     -6.961      0.000      -0.917      -0.514
sigma2        18.6941      1.506     12.413      0.000      15.742      21.646
===================================================================================
Ljung-Box (L1) (Q):                   0.10   Jarque-Bera (JB):               175.68
Prob(Q):                              0.75   Prob(JB):                         0.00
Heteroskedasticity (H):               0.61   Skew:                            -0.71
Prob(H) (two-sided):                  0.06   Kurtosis:                         7.65
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


fcast302= Infmodel_fit302.get_forecast(steps=12)
predicted_inf302 = fcast302.predicted_mean

mse302 = mean_squared_error(test_data, predicted_inf302)
rmse302 = np.sqrt(mse302)

print("Mean Squared Error: ", mse302)
print("Root Mean Squared Error: ", rmse302)

Mean Squared Error:  6.663006329055892
Root Mean Squared Error:  2.5812799788197895


plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted_inf302, label='Predicted Inflation')
plt.title ('ARIMA (3,0,2)')
plt.legend()
plt.show()


fcast302.conf_int()


# Print the forecast values : actual v predicted v lower ci v upper ci (95% confidence interval)

result302 = pd.concat([test_data, predicted_inf302,fcast302.conf_int()], axis=1)

result302 = result302.rename(columns={0: 'Inflation','predicted_mean': 'Pred_Inflation', 'lower Inflation': 'Lower CI',
                                  'upper Inflation': 'Upper CI'})
print(result302)

            Inflation  Pred_Inflation   Lower CI   Upper CI
2022-11-01      13.65        9.873307   1.399072  18.347542
2022-12-01      11.76        8.636863  -6.038946  23.312673
2023-01-01       6.83        9.094397  -9.535271  27.724065
2023-02-01       6.76        8.616126 -14.032250  31.264502
2023-03-01       5.71        8.758568 -17.205575  34.722710
2023-04-01       7.20        8.598580 -20.444135  37.641294
2023-05-01       9.52        8.679874 -22.986068  40.345815
2023-06-01      11.02        8.676722 -25.281053  42.634496
2023-07-01      11.45        8.777236 -27.087565  44.642037
2023-08-01      11.83        8.864589 -28.570273  46.299450
2023-09-01      12.33        9.001903 -29.670988  47.674793
2023-10-01      10.85        9.145340 -30.471275  48.761956


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = result.index.values


upper = result302['Upper CI']
lower = result302['Lower CI']

result302['Inflation'].plot(color = 'blue', label = 'Actual')
result302['Pred_Inflation'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'grey', label = 'Upper CI')
lower.plot(color = 'grey', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)


plt.show()


#viewing modelprediction for plot 
model302 = model302.fit().predict()


#Plotting actual test data and predicted data
plt.plot(NI, label='Actual Inflation')
plt.plot(model302, label='Model predicted Inflation')
plt.plot(predicted_inf, label='Test data Predicted Inflation')
plt.title('ARIMA(3,0,2)')
plt.legend()
plt.show()


# Plot residual errors
residuals101 = pd.DataFrame(Infmodel_fit302.resid)
fig, ax = plt.subplots(1,2)
residuals101.plot(title="Residuals for ARIMA 3,0,2", ax=ax[0],figsize=(15,7))
residuals101.plot(kind='kde',ax=ax[1])
plt.show()


#Plotting all ARIMA predictions against test data 

plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted_inf0, label='Predicted ARIMA 2,0,1')
plt.plot(predicted_inf302, label='Predicted ARIMA 3,0,2')
plt.plot(predicted_inf, label='Predicted ARIMA 3,0,3')
plt.title ('ARIMA Predictions')
plt.legend(loc='lower left',fontsize="15")
plt.show()


from pmdarima.arima import auto_arima


#import pmdarima as pm

model1 = auto_arima(train_data, start_p=1, start_q=1,
                      test='adf',       # use adftest to find optimal 'd'
                      max_p=5, max_q=5, # maximum p and q
                      m=1,              # frequency of series
                      d=False,          
                      seasonal=False,   # No Seasonality
                      start_P=0, 
                      D=0, 
                      trace=True,
                      error_action='ignore',  
                      suppress_warnings=True, 
                      stepwise=True)

print(model1.summary())

Performing stepwise search to minimize aic
 ARIMA(1,False,1)(0,0,0)[0]             : AIC=1072.126, Time=0.03 sec
 ARIMA(0,False,0)(0,0,0)[0]             : AIC=1688.582, Time=0.01 sec
 ARIMA(1,False,0)(0,0,0)[0]             : AIC=1115.573, Time=0.01 sec
 ARIMA(0,False,1)(0,0,0)[0]             : AIC=inf, Time=0.02 sec
 ARIMA(2,False,1)(0,0,0)[0]             : AIC=1061.791, Time=0.02 sec
 ARIMA(2,False,0)(0,0,0)[0]             : AIC=1063.733, Time=0.01 sec
 ARIMA(3,False,1)(0,0,0)[0]             : AIC=1058.339, Time=0.04 sec
 ARIMA(3,False,0)(0,0,0)[0]             : AIC=1064.338, Time=0.02 sec
 ARIMA(4,False,1)(0,0,0)[0]             : AIC=1050.636, Time=0.03 sec
 ARIMA(4,False,0)(0,0,0)[0]             : AIC=1060.373, Time=0.02 sec
 ARIMA(5,False,1)(0,0,0)[0]             : AIC=1052.624, Time=0.07 sec
 ARIMA(4,False,2)(0,0,0)[0]             : AIC=1049.271, Time=0.10 sec
 ARIMA(3,False,2)(0,0,0)[0]             : AIC=1047.401, Time=0.05 sec
 ARIMA(2,False,2)(0,0,0)[0]             : AIC=1056.877, Time=0.03 sec
 ARIMA(3,False,3)(0,0,0)[0]             : AIC=1049.314, Time=0.08 sec
 ARIMA(2,False,3)(0,0,0)[0]             : AIC=1055.242, Time=0.03 sec
 ARIMA(4,False,3)(0,0,0)[0]             : AIC=1051.150, Time=0.05 sec
 ARIMA(3,False,2)(0,0,0)[0] intercept   : AIC=1044.868, Time=0.05 sec
 ARIMA(2,False,2)(0,0,0)[0] intercept   : AIC=1053.134, Time=0.04 sec
 ARIMA(3,False,1)(0,0,0)[0] intercept   : AIC=1058.947, Time=0.05 sec
 ARIMA(4,False,2)(0,0,0)[0] intercept   : AIC=1046.040, Time=0.12 sec
 ARIMA(3,False,3)(0,0,0)[0] intercept   : AIC=1046.410, Time=0.06 sec
 ARIMA(2,False,1)(0,0,0)[0] intercept   : AIC=1060.877, Time=0.03 sec
 ARIMA(2,False,3)(0,0,0)[0] intercept   : AIC=1053.002, Time=0.07 sec
 ARIMA(4,False,1)(0,0,0)[0] intercept   : AIC=1050.660, Time=0.05 sec
 ARIMA(4,False,3)(0,0,0)[0] intercept   : AIC=1047.405, Time=0.10 sec

Best model:  ARIMA(3,False,2)(0,0,0)[0] intercept
Total fit time: 1.174 seconds
                               SARIMAX Results                                
==============================================================================
Dep. Variable:                      y   No. Observations:                  178
Model:               SARIMAX(3, 0, 2)   Log Likelihood                -515.434
Date:                Wed, 01 May 2024   AIC                           1044.868
Time:                        04:16:17   BIC                           1067.140
Sample:                    01-01-2008   HQIC                          1053.900
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
intercept      0.3263      0.199      1.641      0.101      -0.063       0.716
ar.L1          1.3226      0.121     10.927      0.000       1.085       1.560
ar.L2          0.1994      0.205      0.971      0.332      -0.203       0.602
ar.L3         -0.5530      0.100     -5.526      0.000      -0.749      -0.357
ma.L1          0.0914      0.139      0.656      0.512      -0.182       0.365
ma.L2         -0.7152      0.103     -6.961      0.000      -0.917      -0.514
sigma2        18.6939      1.506     12.413      0.000      15.742      21.646
===================================================================================
Ljung-Box (L1) (Q):                   0.10   Jarque-Bera (JB):               175.67
Prob(Q):                              0.75   Prob(JB):                         0.00
Heteroskedasticity (H):               0.61   Skew:                            -0.71
Prob(H) (two-sided):                  0.06   Kurtosis:                         7.65
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


# Forecast

Test_size =12

prediction,conf_int3 = model1.predict(n_periods=len(test_data),return_conf_int=True,alpha=0.05)


mse1= mean_squared_error(test_data, prediction)
rmse1 = np.sqrt(mse1)

print("Mean Squared Error: ", mse1)
print("Root Mean Squared Error: ", rmse1)

Mean Squared Error:  6.657822001602984
Root Mean Squared Error:  2.5802755669894997


#plot the predictions for validation set
#plt.plot(train_data, label='Train')
plt.plot(test_data, label='Test')
plt.plot(prediction, label='Prediction')
plt.title ('Auto ARIMA Prediction')
plt.show()


prediction =pd.DataFrame(prediction,index =test_data.index,columns=["Prediction"])
                          
conf_int3 =pd.DataFrame(conf_int3,index = test_data.index,columns=['LowerCI','UpperCI'])


# Print the forecast values : actual v predicted v lower ci v upper ci (95% confidence interval)

result3 = pd.concat([test_data, prediction,conf_int3], axis=1)

print(result3)

            Inflation  Prediction    LowerCI    UpperCI
date                                                   
2022-11-01      13.65    9.873719   1.399526  18.347912
2022-12-01      11.76    8.637850  -6.038031  23.313730
2023-01-01       6.83    9.095835  -9.534129  27.725800
2023-02-01       6.76    8.617996 -14.030920  31.266912
2023-03-01       5.71    8.760812 -17.204091  34.725715
2023-04-01       7.20    8.601138 -20.442527  37.644802
2023-05-01       9.52    8.682693 -22.984335  40.349720
2023-06-01      11.02    8.679734 -25.279211  42.638679
2023-07-01      11.45    8.780386 -27.085613  44.646385
2023-08-01      11.83    8.867813 -28.568224  46.303849
2023-09-01      12.33    9.005144 -29.668854  47.679142
2023-10-01      10.85    9.148542 -30.469082  48.766166


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = result3.index.values


upper = result3['UpperCI']
lower = result3['LowerCI']

result3['Inflation'].plot(color = 'blue', label = 'Actual')
result3['Prediction'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'Brown', label = 'Upper CI')
lower.plot(color = 'Brown', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)
plt.title('Auto ARIMA Actual v Prediction v Confidence Intervals')

plt.show()


#viewing modelprediction for plot 
modelAuto = model1.fittedvalues()


#Plotting actual test data and predicted data
plt.plot(train_data, label='Actual Inflation')
plt.plot(modelAuto, label='Auto ARIMA Model fitted values')
#plt.plot(prediction, label='Test data Predicted Inflation')
plt.title('Auto ARIMA')
plt.legend()
plt.show()


#residue auto arima
model1.plot_diagnostics(figsize=(20,15))
plt.show()


from statsmodels.tsa.api import SimpleExpSmoothing


#First Instance
#best pratice alpha between 0.1 - 1.0

ins1 = SimpleExpSmoothing(train_data).fit(smoothing_level=0.2,optimized=False)
ins_cast1 = ins1.forecast(12).rename('alpha=0.2')

#Second Instance
ins2 = SimpleExpSmoothing(train_data).fit(smoothing_level=0.8,optimized=False)
ins_cast2 = ins2.forecast(12).rename('alpha=0.8')

#Third Instance
ins3 = SimpleExpSmoothing(train_data).fit(smoothing_level=0.99,optimized=False)
ins_cast3 = ins3.forecast(12).rename('alpha=0.99')


resultSES = pd.concat([ins_cast1,ins_cast2,ins_cast3], axis=1)

print(resultSES)

            alpha=0.2  alpha=0.8  alpha=0.99
2022-11-01  10.593457   7.382923    8.311086
2022-12-01  10.593457   7.382923    8.311086
2023-01-01  10.593457   7.382923    8.311086
2023-02-01  10.593457   7.382923    8.311086
2023-03-01  10.593457   7.382923    8.311086
2023-04-01  10.593457   7.382923    8.311086
2023-05-01  10.593457   7.382923    8.311086
2023-06-01  10.593457   7.382923    8.311086
2023-07-01  10.593457   7.382923    8.311086
2023-08-01  10.593457   7.382923    8.311086
2023-09-01  10.593457   7.382923    8.311086
2023-10-01  10.593457   7.382923    8.311086


#Plotting all SES predictions against test data 

plt.plot(test_data, label='Actual Inflation')
plt.plot(ins_cast1, label='SES 0.2')
plt.plot(ins_cast2, label='SES 0.8')
plt.plot(ins_cast3, label='SES 0.99')
plt.title ('Simple exponential smoothing')
plt.legend()
plt.show()


#SES model evaluation

print('SES Instance 1')
sesmse1 = ((ins_cast1 - test_data) ** 2).mean()
print('Mean Squared Error',sesmse1)
print('The Root Mean Squared Error for instance 1 {}'.format(round(np.sqrt(sesmse1), 2)))
print('')

print('SES Instance 2')
sesmse2 = ((ins_cast2 - test_data) ** 2).mean()
print('Mean Squared Error',sesmse2)
print('The Root Mean Squared Error for instance 2 {}'.format(round(np.sqrt(sesmse2), 2)))
print('')
print('SES Instance 3')
sesmse3 = ((ins_cast3 - test_data) ** 2).mean()
print('Mean Squared Error',sesmse3)
print('The Root Mean Squared Error for instance 3 {}'.format(round(np.sqrt(sesmse3), 2)))

SES Instance 1
Mean Squared Error 6.800361410055941
The Root Mean Squared Error for instance 1 2.61

SES Instance 2
Mean Squared Error 12.714012782597047
The Root Mean Squared Error for instance 2 3.57

SES Instance 3
Mean Squared Error 8.885968653689853
The Root Mean Squared Error for instance 3 2.98


#creating MA using ARMA model by defining q only q=2

ARMA_model = ARIMA(train_data, order=(0, 0,2))
results = ARMA_model.fit()
print(results.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                 ARIMA(0, 0, 2)   Log Likelihood                -647.616
Date:                Wed, 01 May 2024   AIC                           1303.233
Time:                        04:17:24   BIC                           1315.960
Sample:                    01-01-2008   HQIC                          1308.394
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         11.2090      2.668      4.201      0.000       5.980      16.438
ma.L1          1.5409      0.054     28.603      0.000       1.435       1.646
ma.L2          0.7625      0.056     13.636      0.000       0.653       0.872
sigma2        83.1638      9.135      9.104      0.000      65.259     101.069
===================================================================================
Ljung-Box (L1) (Q):                  38.66   Jarque-Bera (JB):                22.26
Prob(Q):                              0.00   Prob(JB):                         0.00
Heteroskedasticity (H):               1.39   Skew:                             0.78
Prob(H) (two-sided):                  0.20   Kurtosis:                         3.75
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


#viewing modelprediction for plot 
ARMA_modelpred = ARMA_model.fit().predict()


## q = 3


#creating MA using ARMA model by defining q only q=3

ARMA_model1 = ARIMA(train_data, order=(0, 0,3))
result1 = ARMA_model1.fit()
print(result1.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                 ARIMA(0, 0, 3)   Log Likelihood                -594.402
Date:                Wed, 01 May 2024   AIC                           1198.804
Time:                        04:17:34   BIC                           1214.713
Sample:                    01-01-2008   HQIC                          1205.256
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         11.4150      2.901      3.934      0.000       5.728      17.102
ma.L1          1.8010      0.055     32.908      0.000       1.694       1.908
ma.L2          1.6488      0.085     19.419      0.000       1.482       1.815
ma.L3          0.7796      0.055     14.251      0.000       0.672       0.887
sigma2        45.3081      4.192     10.808      0.000      37.092      53.525
===================================================================================
Ljung-Box (L1) (Q):                   9.01   Jarque-Bera (JB):                 8.24
Prob(Q):                              0.00   Prob(JB):                         0.02
Heteroskedasticity (H):               0.91   Skew:                             0.33
Prob(H) (two-sided):                  0.72   Kurtosis:                         3.83
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).


#viewing modelprediction for plot 
ARMA_modelpred1 = ARMA_model1.fit().predict()


#Plotting actual train data, model 
plt.plot(train_data, label='Actual Inflation')
plt.plot(ARMA_modelpred , label='2_MA')
plt.title('Simple moving average')
plt.legend()
plt.show()


#Plotting actual train data, model 
plt.plot(train_data, label='Actual Inflation')
plt.plot(ARMA_modelpred1 , label='3_MA')
plt.title('Simple moving average')
plt.legend()
plt.show()


## Evaluation of ARMA(0,0,2)


ARMAfcast= results.get_forecast(steps=12)
predicted002 = ARMAfcast.predicted_mean


predicted002=predicted002.rename('2MA')


mse002 = mean_squared_error(test_data, predicted002)
rmse002 = np.sqrt(mse002)

print("Mean Squared Error: ", mse002)
print("Root Mean Squared Error: ", rmse002)

Mean Squared Error:  7.750802143259662
Root Mean Squared Error:  2.7840262468697494


## Evaluation of ARMA(0,0,3)


ARMAfcast3= result1.get_forecast(steps=12)
predicted003 = ARMAfcast3.predicted_mean


predicted003= predicted003.rename('3MA')


mse003 = mean_squared_error(test_data, predicted003)
rmse003 = np.sqrt(mse003)

print("Mean Squared Error: ", mse003)
print("Root Mean Squared Error: ", rmse003)

Mean Squared Error:  10.476113383149054
Root Mean Squared Error:  3.236682465604103


#q=12


#creating MA using ARMA model by defining q only q=12

ARMA_model12 = ARIMA(train_data, order=(0, 0,12))
result12 = ARMA_model12.fit()
print(result12.summary())

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  178
Model:                ARIMA(0, 0, 12)   Log Likelihood                -495.586
Date:                Wed, 01 May 2024   AIC                           1019.173
Time:                        04:17:50   BIC                           1063.718
Sample:                    01-01-2008   HQIC                          1037.237
                         - 10-01-2022                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         11.2970      5.825      1.940      0.052      -0.119      22.713
ma.L1          1.4265      1.561      0.914      0.361      -1.634       4.487
ma.L2          1.4253      3.049      0.468      0.640      -4.550       7.400
ma.L3          1.5071      3.246      0.464      0.642      -4.856       7.870
ma.L4          1.4087      2.255      0.625      0.532      -3.011       5.828
ma.L5          1.5065      0.201      7.476      0.000       1.112       1.901
ma.L6          1.5561      2.461      0.632      0.527      -3.268       6.380
ma.L7          1.6619      4.565      0.364      0.716      -7.286      10.609
ma.L8          1.6365      5.581      0.293      0.769      -9.302      12.575
ma.L9          1.3726      5.475      0.251      0.802      -9.359      12.104
ma.L10         1.2671      4.654      0.272      0.785      -7.855      10.389
ma.L11         1.1673      2.999      0.389      0.697      -4.711       7.045
ma.L12         0.5009      0.949      0.528      0.598      -1.359       2.361
sigma2        14.2119     26.365      0.539      0.590     -37.463      65.887
===================================================================================
Ljung-Box (L1) (Q):                   0.15   Jarque-Bera (JB):               179.92
Prob(Q):                              0.70   Prob(JB):                         0.00
Heteroskedasticity (H):               0.69   Skew:                            -0.25
Prob(H) (two-sided):                  0.16   Kurtosis:                         7.90
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/base/model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "


#viewing modelprediction for plot 
ARMA_modelpred12 = ARMA_model12.fit().predict()

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/base/model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "


#Plotting actual train data, model 
plt.plot(train_data, label='Actual Inflation')
plt.plot(ARMA_modelpred12 , label='12_MA Model fitted values')
plt.title('Simple moving average')
plt.legend()
plt.show()


#Prediction and Evaluation q=12


ARMAfcast12= result12.get_forecast(steps=12)
predicted12 = ARMAfcast12.predicted_mean


predicted12=predicted12.rename('12MA')


predicted12

2022-11-01    10.071227
2022-12-01     7.694820
2023-01-01     6.147343
2023-02-01     4.498324
2023-03-01     5.271082
2023-04-01     8.600758
2023-05-01    12.020680
2023-06-01    12.905037
2023-07-01    13.109064
2023-08-01    14.746046
2023-09-01    15.773707
2023-10-01    13.469375
Freq: MS, Name: 12MA, dtype: float64


mse12 = mean_squared_error(test_data, predicted12)
rmse12 = np.sqrt(mse12)

print("Mean Squared Error: ", mse12)
print("Root Mean Squared Error: ", rmse12)

Mean Squared Error:  6.404341805136163
Root Mean Squared Error:  2.5306801072312877


## best SMA model q=12


#concating test data, predicted inflation and confidence intervals

result012 = pd.concat([test_data, predicted12,ARMAfcast12.conf_int()]
                      , axis=1)


#Renaming columns
result012 = result012.rename(columns={0: 'Inflation','12MA': 
                'Pred_Inflation', 'lower Inflation': 'Lower CI',
                                  'upper Inflation': 'Upper CI'})
print(result012)

            Inflation  Pred_Inflation   Lower CI   Upper CI
2022-11-01      13.65       10.071227   2.629446  17.513009
2022-12-01      11.76        7.694820  -5.321434  20.711073
2023-01-01       6.83        6.147343 -10.702067  22.996754
2023-02-01       6.76        4.498324 -15.790763  24.787412
2023-03-01       5.71        5.271082 -17.601217  28.143381
2023-04-01       7.20        8.600758 -16.871848  34.073363
2023-05-01       9.52       12.020680 -15.909604  39.950963
2023-06-01      11.02       12.905037 -17.548355  43.358428
2023-07-01      11.45       13.109064 -19.596485  45.814613
2023-08-01      11.83       14.746046 -19.450300  48.942393
2023-09-01      12.33       15.773707 -19.643211  51.190626
2023-10-01      10.85       13.469375 -22.960316  49.899067


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = result012.index.values


upper = result012['Upper CI']
lower = result012['Lower CI']

result012['Inflation'].plot(color = 'blue', label = 'Actual')
result012['Pred_Inflation'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'grey', label = 'Upper CI')
lower.plot(color = 'grey', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)
plt.title('12MA Actual v Prediction v Confidence Intervals')

plt.show()


# Plot residual errors
residuals0012 = pd.DataFrame(result12.resid) 
fig, ax = plt.subplots(1,2)
residuals101.plot(title="Residuals for 12MA ", ax=ax[0],figsize=(15,7))
residuals101.plot(kind='kde',ax=ax[1])
plt.show()


##concat all SMA models; 2ma,3ma,12ma


MA = pd.concat([test_data,predicted002,predicted003,predicted12],axis=1)
print(MA)

            Inflation        2MA        3MA       12MA
2022-11-01      13.65  13.718823  13.981716  10.071227
2022-12-01      11.76  13.493918  14.613581   7.694820
2023-01-01       6.83  11.209011  13.186178   6.147343
2023-02-01       6.76  11.209011  11.415005   4.498324
2023-03-01       5.71  11.209011  11.415005   5.271082
2023-04-01       7.20  11.209011  11.415005   8.600758
2023-05-01       9.52  11.209011  11.415005  12.020680
2023-06-01      11.02  11.209011  11.415005  12.905037
2023-07-01      11.45  11.209011  11.415005  13.109064
2023-08-01      11.83  11.209011  11.415005  14.746046
2023-09-01      12.33  11.209011  11.415005  15.773707
2023-10-01      10.85  11.209011  11.415005  13.469375


#Plotting actual test data, predictions 
plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted002, label='2_MA')
plt.plot(predicted003, label='3_MA')
plt.plot(predicted12, label='12_MA') 
plt.title('Simple moving average')
plt.legend()
plt.show()


train_len = 178


train = pd.DataFrame(train_data)
test = pd.DataFrame(test_data)


y_hat_naive = test.copy()
y_hat_naive['naive_forecast'] = train['Inflation'][train_len-1]


plt.figure(figsize=(10,5))
#plt.plot(train['Inflation'], label='Train')
plt.plot(test['Inflation'], label='Test')
plt.plot(y_hat_naive['naive_forecast'], label='Naive forecast')
plt.legend(loc='best')
plt.title('Naive Method')
plt.show()


y_hat_naive


mse_N = mean_squared_error(test['Inflation'], y_hat_naive['naive_forecast']).round(2)
rmse_N = np.sqrt(mean_squared_error(test['Inflation'], y_hat_naive['naive_forecast'])).round(2)

print('MSE', mse_N)
print('RMSE',rmse_N)

MSE 8.73
RMSE 2.96


## Best Model consideration


#Plotting actual test data, predictions 
plt.plot(test_data, label='Actual Inflation')
plt.plot(predicted12, label='12_MA') 
plt.plot(predicted_inf302 , label='ARIMA(3,0,2)') 
plt.plot(prediction, label='Auto ARIMA') 
plt.title('Best Models')
plt.legend()
plt.show()


## Chosen model to forecast next 12 months ARIMA (3,0,2)


#retrained model with full data to predict next 12 months 

Finalmodel = ARIMA(NI,order=(0,0,12))
# Training arima modeling 
Infmodel_fitFinal = Finalmodel.fit()
print(Infmodel_fitFinal.summary())

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/tsa/base/tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
/opt/anaconda3/lib/python3.9/site-packages/statsmodels/tsa/base/tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)
/opt/anaconda3/lib/python3.9/site-packages/statsmodels/tsa/base/tsa_model.py:471: ValueWarning: No frequency information was provided, so inferred frequency MS will be used.
  self._init_dates(dates, freq)

                               SARIMAX Results                                
==============================================================================
Dep. Variable:              Inflation   No. Observations:                  190
Model:                ARIMA(0, 0, 12)   Log Likelihood                -564.383
Date:                Wed, 01 May 2024   AIC                           1156.766
Time:                        04:18:53   BIC                           1202.224
Sample:                    01-01-2008   HQIC                          1175.180
                         - 10-01-2023                                         
Covariance Type:                  opg                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const         11.2107      8.035      1.395      0.163      -4.538      26.959
ma.L1          1.7524      0.367      4.776      0.000       1.033       2.471
ma.L2          2.3177      0.666      3.479      0.001       1.012       3.624
ma.L3          2.5389      0.648      3.918      0.000       1.269       3.809
ma.L4          2.4854      0.411      6.051      0.000       1.680       3.290
ma.L5          2.5026      1.080      2.318      0.020       0.387       4.619
ma.L6          2.5567      1.730      1.478      0.139      -0.834       5.948
ma.L7          2.4619      1.820      1.353      0.176      -1.104       6.028
ma.L8          1.9282      1.346      1.432      0.152      -0.711       4.567
ma.L9          0.9607      0.728      1.320      0.187      -0.466       2.387
ma.L10         0.2308      0.339      0.680      0.496      -0.434       0.896
ma.L11         0.1523      0.192      0.793      0.428      -0.224       0.529
ma.L12        -0.1414      0.099     -1.423      0.155      -0.336       0.053
sigma2        20.2551      9.122      2.221      0.026       2.377      38.133
===================================================================================
Ljung-Box (L1) (Q):                   0.54   Jarque-Bera (JB):                15.04
Prob(Q):                              0.46   Prob(JB):                         0.00
Heteroskedasticity (H):               1.27   Skew:                             0.13
Prob(H) (two-sided):                  0.35   Kurtosis:                         4.35
===================================================================================

Warnings:
[1] Covariance matrix calculated using the outer product of gradients (complex-step).

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/base/model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "


finalfcast= Infmodel_fitFinal.get_forecast(steps=12)
predicted_infFinal = finalfcast.predicted_mean


predpred=Finalmodel.fit().predict()

/opt/anaconda3/lib/python3.9/site-packages/statsmodels/base/model.py:604: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
  warnings.warn("Maximum Likelihood optimization failed to "


predicted_infFinal

2023-11-01     2.033087
2023-12-01    -7.145871
2024-01-01   -13.732559
2024-02-01   -17.208670
2024-03-01   -14.007895
2024-04-01    -2.898789
2024-05-01     9.557748
2024-06-01    15.515285
2024-07-01    14.896806
2024-08-01    13.010454
2024-09-01    12.293819
2024-10-01    10.532069
Freq: MS, Name: predicted_mean, dtype: float64


#Plotting Forecast next 12 months

plt.plot(predicted_infFinal, label='Predicted')
plt.title ('Forecast next 12 months')
plt.legend()
plt.show()


finalresult = pd.concat([predicted_infFinal,finalfcast.conf_int()]
                      , axis=1)


finalresult


#Plotting all ARIMA predictions against test data 

plt.plot(NI, label='Actual Inflation')
plt.plot(predicted_infFinal, label='Predicted')
plt.title ('Forecast next 12 months')
plt.legend()
plt.show()


plt.figure(figsize=(20,10))
plt.plot(NI, label="Actual")
plt.plot(predpred, label="Model fitted values")
plt.title('Nigerian Food Inflation', fontsize=20)
plt.legend()

<matplotlib.legend.Legend at 0x147dfbd60>


# make a plot of model fit
# Narrow confidence interval better precision

fig = plt.figure(figsize = (16,8))
ax1 = fig.add_subplot(111)


x = finalresult.index.values


upper = finalresult['upper Inflation']
lower = finalresult['lower Inflation']

#finalresult['Inflation'].plot(color = 'blue', label = 'Actual')
finalresult['predicted_mean'].plot(color = 'orange',label = 'Predicted' )
upper.plot(color = 'grey', label = 'Upper CI')
lower.plot(color = 'grey', label = 'Lower CI')

# plot the legend for the first plot
plt.legend(loc = 'lower left', fontsize = 12)


# fill between the conf intervals
plt.fill_between(x, lower, upper, color='grey', alpha= 0.2)
plt.title('12MA Actual v Prediction v Confidence Intervals')

plt.show()


## End

	Open	High	Low	Close	Inflation	country	ISO3	date
0	0.53	0.54	0.53	0.53	NaN	Afghanistan	AFG	2007-01-01
1	0.53	0.54	0.53	0.53	NaN	Afghanistan	AFG	2007-02-01
2	0.54	0.54	0.53	0.53	NaN	Afghanistan	AFG	2007-03-01
3	0.53	0.55	0.53	0.55	NaN	Afghanistan	AFG	2007-04-01
4	0.56	0.57	0.56	0.57	NaN	Afghanistan	AFG	2007-05-01
...	...	...	...	...	...	...	...	...
4793	2.74	2.78	2.70	2.75	-0.28	Yemen, Rep.	YEM	2023-06-01
4794	2.79	2.83	2.75	2.81	-1.85	Yemen, Rep.	YEM	2023-07-01
4795	2.85	2.89	2.81	2.83	-3.17	Yemen, Rep.	YEM	2023-08-01
4796	2.86	2.97	2.82	2.97	1.68	Yemen, Rep.	YEM	2023-09-01
4797	3.06	3.11	2.98	2.98	3.76	Yemen, Rep.	YEM	2023-10-01

	Open	High	Low	Close	Inflation
count	190.000000	190.000000	190.000000	190.000000	190.000000
mean	0.945263	0.968842	0.922579	0.945105	11.210526
std	0.403294	0.412335	0.395182	0.403276	24.472839
min	0.550000	0.560000	0.540000	0.550000	-22.710000
25%	0.642500	0.660000	0.622500	0.640000	-5.190000
50%	0.760000	0.785000	0.750000	0.760000	3.765000
75%	1.110000	1.145000	1.072500	1.085000	23.497500
max	2.020000	2.060000	1.980000	1.980000	77.390000

	Open	High	Low	Close	Inflation	year	month
count	190.000000	190.000000	190.000000	190.000000	190.000000	190.000000	190.000000
mean	0.945263	0.968842	0.922579	0.945105	11.210526	2015.421053	6.447368
std	0.403294	0.412335	0.395182	0.403276	24.472839	4.581026	3.440326
min	0.550000	0.560000	0.540000	0.550000	-22.710000	2008.000000	1.000000
25%	0.642500	0.660000	0.622500	0.640000	-5.190000	2011.250000	3.250000
50%	0.760000	0.785000	0.750000	0.760000	3.765000	2015.000000	6.000000
75%	1.110000	1.145000	1.072500	1.085000	23.497500	2019.000000	9.000000
max	2.020000	2.060000	1.980000	1.980000	77.390000	2023.000000	12.000000

	Close_x	Inflation_x
Close_y	1.0000	0.0
Inflation_y	0.0139	1.0

Data preparation¶

EDA¶

Feature selection and testing¶

Trend, Seasonality,Stationarity¶

Model building¶

ARIMA 3,0,3 Model¶

ARIMA 2,0,1 Model¶

ARIMA(3,0,2)¶

Plotting All Manual ARIMA Models¶

AUTO ARIMA model¶

Exponential Smoothing¶

Simple moving Average¶

Naive method¶

	Open	High	Low	Close	Inflation	country	ISO3	date
3456	0.51	0.51	0.49	0.51	NaN	Nigeria	NGA	2007-01-01
3457	0.50	0.51	0.47	0.47	NaN	Nigeria	NGA	2007-02-01
3458	0.45	0.47	0.44	0.47	NaN	Nigeria	NGA	2007-03-01
3459	0.47	0.48	0.46	0.46	NaN	Nigeria	NGA	2007-04-01
3460	0.46	0.47	0.45	0.46	NaN	Nigeria	NGA	2007-05-01
3461	0.46	0.47	0.45	0.46	NaN	Nigeria	NGA	2007-06-01
3462	0.46	0.47	0.45	0.46	NaN	Nigeria	NGA	2007-07-01
3463	0.46	0.47	0.46	0.47	NaN	Nigeria	NGA	2007-08-01
3464	0.48	0.49	0.47	0.48	NaN	Nigeria	NGA	2007-09-01
3465	0.48	0.52	0.47	0.52	NaN	Nigeria	NGA	2007-10-01
3466	0.55	0.57	0.53	0.57	NaN	Nigeria	NGA	2007-11-01
3467	0.60	0.61	0.58	0.58	NaN	Nigeria	NGA	2007-12-01

	Open	High	Low	Close	Inflation	country	ISO3	date
3468	0.58	0.61	0.57	0.61	21.11	Nigeria	NGA	2008-01-01
3469	0.64	0.66	0.62	0.62	32.49	Nigeria	NGA	2008-02-01
3470	0.63	0.64	0.61	0.62	32.65	Nigeria	NGA	2008-03-01
3471	0.62	0.63	0.61	0.61	32.94	Nigeria	NGA	2008-04-01
3472	0.61	0.62	0.59	0.62	34.75	Nigeria	NGA	2008-05-01
...	...	...	...	...	...	...	...	...
3653	1.86	1.92	1.82	1.92	11.02	Nigeria	NGA	2023-06-01
3654	1.96	2.00	1.92	1.98	11.45	Nigeria	NGA	2023-07-01
3655	2.02	2.06	1.98	1.98	11.83	Nigeria	NGA	2023-08-01
3656	1.99	2.03	1.95	1.95	12.33	Nigeria	NGA	2023-09-01
3657	1.94	1.97	1.90	1.90	10.85	Nigeria	NGA	2023-10-01

	lower Inflation	upper Inflation
2022-11-01	1.136343	18.062214
2022-12-01	-6.832552	22.623234
2023-01-01	-11.022703	26.424182
2023-02-01	-15.920565	29.239382
2023-03-01	-19.352330	32.030610
2023-04-01	-22.834247	34.261586
2023-05-01	-25.466755	36.398314
2023-06-01	-27.861379	38.154459
2023-07-01	-29.657317	39.759132
2023-08-01	-31.123331	41.082691
2023-09-01	-32.131452	42.245646
2023-10-01	-32.819296	43.201996

	lower Inflation	upper Inflation
2022-11-01	0.551471	18.513711
2022-12-01	-3.773209	24.961142
2023-01-01	-7.774027	30.835810
2023-02-01	-11.539552	36.209285
2023-03-01	-15.038406	41.041937
2023-04-01	-18.228879	45.291706
2023-05-01	-21.080193	48.934409
2023-06-01	-23.577026	51.967266
2023-07-01	-25.719426	54.407768
2023-08-01	-27.521134	56.290979
2023-09-01	-29.007275	57.666208
2023-10-01	-30.211755	58.593462

	lower Inflation	upper Inflation
2022-11-01	1.399072	18.347542
2022-12-01	-6.038946	23.312673
2023-01-01	-9.535271	27.724065
2023-02-01	-14.032250	31.264502
2023-03-01	-17.205575	34.722710
2023-04-01	-20.444135	37.641294
2023-05-01	-22.986068	40.345815
2023-06-01	-25.281053	42.634496
2023-07-01	-27.087565	44.642037
2023-08-01	-28.570273	46.299450
2023-09-01	-29.670988	47.674793
2023-10-01	-30.471275	48.761956

	Inflation	naive_forecast
date
2022-11-01	13.65	8.36
2022-12-01	11.76	8.36
2023-01-01	6.83	8.36
2023-02-01	6.76	8.36
2023-03-01	5.71	8.36
2023-04-01	7.20	8.36
2023-05-01	9.52	8.36
2023-06-01	11.02	8.36
2023-07-01	11.45	8.36
2023-08-01	11.83	8.36
2023-09-01	12.33	8.36
2023-10-01	10.85	8.36

	predicted_mean	lower Inflation	upper Inflation
2023-11-01	2.033087	-6.872632	10.938807
2023-12-01	-7.145871	-25.130208	10.838466
2024-01-01	-13.732559	-41.125204	13.660087
2024-02-01	-17.208670	-52.709229	18.291889
2024-03-01	-14.007895	-55.781541	27.765751
2024-04-01	-2.898789	-50.113089	44.315510
2024-05-01	9.557748	-42.695749	61.811246
2024-06-01	15.515285	-40.983440	72.014010
2024-07-01	14.896806	-44.055873	73.849485
2024-08-01	13.010454	-46.535436	72.556344
2024-09-01	12.293819	-47.286036	71.873675
2024-10-01	10.532069	-49.062702	70.126839